1.Introduction

In this paper, we theoretically determine the temperature distribution of a
homogeneous fluid-saturated porous medium due to the injection of hot water through
a perforated pipe embedded in the medium. Physically, this is a way of heating the
porous matrix and the fluids therein. The problem is ubiquitous in technological
areas like heavy and extra-heavy oil recovery ^{1}^{-}^{6} and for the production and storage of energy in geothermal
systems ^{7}^{-}^{12}, among others.

The injection of hot water into a porous medium does imply the simultaneous inflow of
mass and thermal energy (enthalpy), both by conduction and advection. In the
theoretical treatments, researchers commonly have treated the injection problem as
one where the temperature of the injected fluid and the initial temperature of the
fluid-saturated porous medium are known, while the spatial and temporal profiles of
the temperature of the medium around the injection pipe are computed ^{1}^{-}^{11}.

In the current work, we assume that the hot water has penetrated the porous medium a
distance large compared to the radius of the injection pipe, which is then idealized
as a line source of mass and heat ^{13}. However, we neglect the effects of gravity, which would
come into play at later times. This leaves aside buoyancy-driven flows in miscible
and immiscible fluids ^{14}. Such
cases correspond, for instance, to the early stages of the injection of hot water
into heavy and extra-heavy oil reservoirs (immiscible fluids) and to the injection
of hot or cold water into geothermal aquifers (miscible fluids). In these cases, the
densities of the fluids involved have similar values, and thermal convection can be
assumed to play a secondary role. We also neglect finger-like instabilities.

The basis for the study of problems of hot fluid injection is the energy conservation equation for fluid-saturated porous media. For transient problems, this is a partial differential equation for the temperature as a function of time and the spatial coordinates, which involves advection and conduction heat transfer and initial and boundary conditions that depend on the specific problem considered.

Hot water flooding is a thermal method used in petroleum engineering. It is useful to
enhance oil recovery because the high viscosity of the *in situ* oil
drastically decreases when its temperature is increased ^{1}^{-}^{6}. In the case of geothermal systems, cold or hot water
can be injected depending on the purpose. Cold water is injected to extract heat
from the hot rock by having the water heated before the waterfront reaches the
production wells ^{7}^{-}^{11}. Hot water is pumped into shallow
permeable layers of rock for seasonal heat storage into reservoirs ^{12}.

In the models, commonly, rock and fluid properties such as the specific heat, density, and thermal conductivity, are considered to be constant in the reservoir. This assumption is valid when the change of temperature in the porous media is small.

The objective of this study is to understand the heat transfer mechanisms in homogeneous porous media through the theoretical modeling of heat transfer coupled with fluid flow. The paper is organized as follows. The physical ingredients of the problem are reviewed in Sec. 2. The mathematical problem is formulated and solved in Sec. 3, showing that the temperature distribution is self-similar and depends on a single dimensionless parameter, a Peclet number that measures the ratio of advection to conduction heat transfer. The features of the temperature profile for slow, medium, and fast injection are discussed. Finally, Sec. 4 summarizes the main conclusions of the work.

2. Physical model

The temperature field around the injection pipe will be analyzed to arrive at conclusions of practical interest.

The hot water to be injected flows through a pipe immersed in the porous medium and leaves the pipe through an array of uniformly distributed orifices in the pipe wall. Owing to the low viscosity of the water, the pressure drop in the pipe can be neglected so that the injection pressure is uniform along the pipe and the flow ensuing in the porous medium can be taken to be two-dimensional in planes perpendicular to the pipe, and purely radial at distances from the pipe large compared to its radius.

The porous medium is homogeneous, with permeability *K* and porosity
φ, and is initially saturated with a quiescent liquid of density nearly equal to the
density p of the water and temperature _{w}. For example, this liquid could be a heavy or extra-heavy oil or
fresh/saline water, all of which have densities very similar to the density of the
hot water. The liquid is assumed to be immiscible with the injected water at the
time scale of the process. We also assume that the fluid and solid phases of the
porous medium have the same local temperature (local thermal equilibrium. The
pressure in the pipe is such that a constant flow rate of water, *q*
per unit length of the pipe, is injected.

The injected water cools down while moving radially out by transferring heat to the
solid matrix and surrounding liquid. The density change of the water during this
process is small so that the approximation of constant *p* is
applicable.

3.Analytical model

Figure 1 is a sketch of the problem with the
idealizations described in the previous paragraphs. With these idealizations, the
radial filtration velocity in the porous medium is

from the condition

The temperature of the porous medium obeys the energy equation ^{10}^{,}^{15}^{,}^{16}

where

These magnitudes will, in general, be discontinuous at

Defining

and writing (2) in conservative form, we have

where

The quantity ^{17}. Since typically
_{
f
}^{17}^{,}^{18}.

Equation (5) must be solved with the initial and boundary conditions

where ^{19}.

The solution of (5) and (6) is self-similar, of the form

Carrying this to (5) and (6), we find

where

The solution of (8) is

which is plotted in Fig. 2 for three values of
the Peclet number Pe. As can be seen, _{w} of the water in
the injection pipe:

The figure shows three very different behaviors for the quantity

The importance of the dimensionless plot in Fig.
2 is that it shows different ways of cooling down the injected hot fluid
in the porous medium, depending primarily on the value of the Peclet number. For
instance, a low value of Pe (which can be given as

In the context of oil recovery, sand (unconsolidated) ^{1}
^{2} and sandstone (consolidated)
^{4} reservoirs, which store large
reserves of heavy and extra-heavy oil, have been exploited by using hot water
injection. In these cases, the Peclet number can be determined using
order-of-magnitude estimates as follows: the average thermal conductivity k_{
m
} is, in both cases, in the order of 1 W/m K in a wide range of temperatures
^{20}^{,}^{21}, meaning that the heat transfer
from the water to the porous matrix is very efficient. The volumetric heat capacity
of water ^{3}K)^{22}). Consequently, the ratio ^{6}s/m^{2}, and it is found that for
water flooding into this type of reservoir, the best form of change Pe is by
changing the injection rate. As an example, if a process requires ^{2}/s, and so forth and so on for the other Peclet numbers.
Geothermal aquifers have similar values of the ratio ^{23} and, therefore, similar
conclusions as aforementioned for Pe and q are reached.

4.Conclusions

An analysis has been carried out of the heating of an unbounded porous medium by injection of hot water through a perforated pipe. After a short initial period that is not analyzed, the size of the heated region becomes large compared to the radius of the injection pipe, which can be idealized as a line source of mass and heat. On the other hand, the temperature distribution is axisymmetric before buoyancy comes into play. When both conditions are satisfied, the solution is self-similar and depends only on a Peclet number. This solution has been computed, and its dependence on the Peclet number has been discussed. In conditions typical of hot water injection into the sand and sandstone heavy oil reservoirs, the ratio of volumetric heat capacity of water to the average thermal conductivity of the medium is very large, and the Peclet number is essentially a measure of the flow rate of water injected, which determines the way the thermal energy behaves during the injection.